A 1-arc-regular graph, also called a 1-regular graph, is a graph whose automorphism group acts regularly
on its arcs . Thus, for every ordered pair of arcs, exactly
one graph automorphism maps the first arc to
the second.
For connected cubic 1-arc-regular graphs, the line graph construction gives a one-to-one correspondence with quartic half-arc-transitive graphs of girth
3 (Marušič and Xu 1997).
See also Arc-Transitive Graph ,
Graph Arc ,
Regular Group
Action
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References Marušič, D. and Xu, M. Y. "A 1/2-Transitive Graph of Valency 4 with a Nonsolvable Group of Automorphisms." J. Graph Th. 25 ,
133-138, 1997. https://doi.org/10.1002/(SICI)1097-0118(199706)25:2%3C133::AID-JGT5%3E3.0.CO;2-N .
Cite this as:
Weisstein, Eric W. "1-Arc-Regular Graph."
From MathWorld https://mathworld.wolfram.com/1-Arc-RegularGraph.html
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